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semieunit: moved doc location, fix to IsomorphismSemigroups
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This commit moves the documentation for McAlister triple semigroups to
their own chapter - Chapter 12. It also re-implements
IsomorphismSemigroups, which previously was not working, and adds more
tests for it.
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ChristopherRussell committed Aug 28, 2017
1 parent cf7b780 commit 753090d
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19 changes: 10 additions & 9 deletions doc/main.xml
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<#Include SYSTEM "z-chap09.xml"> <!-- standard constructions-->
<#Include SYSTEM "z-chap10.xml"> <!-- free objects -->
<#Include SYSTEM "z-chap11.xml"> <!-- graph inverse semigroups -->
<#Include SYSTEM "z-chap12.xml"> <!-- Green's relations -->
<#Include SYSTEM "z-chap13.xml"> <!-- attributes of semigroups -->
<#Include SYSTEM "z-chap14.xml"> <!-- properties of semigroups -->
<#Include SYSTEM "z-chap15.xml"> <!-- properties and attributes of inverse semigroups -->
<#Include SYSTEM "z-chap16.xml"> <!-- congruences -->
<#Include SYSTEM "z-chap17.xml"> <!-- homomorphisms -->
<#Include SYSTEM "z-chap18.xml"> <!-- visualising semigroups -->
<#Include SYSTEM "z-chap19.xml"> <!-- utils -->
<!--<#Include SYSTEM "z-chap20.xml"> <!-- orbits -->
<#Include SYSTEM "z-chap12.xml"> <!-- McAlister triple semigroups -->
<#Include SYSTEM "z-chap13.xml"> <!-- Green's relations -->
<#Include SYSTEM "z-chap14.xml"> <!-- attributes of semigroups -->
<#Include SYSTEM "z-chap15.xml"> <!-- properties of semigroups -->
<#Include SYSTEM "z-chap16.xml"> <!-- properties and attributes of inverse semigroups -->
<#Include SYSTEM "z-chap17.xml"> <!-- congruences -->
<#Include SYSTEM "z-chap18.xml"> <!-- homomorphisms -->
<#Include SYSTEM "z-chap19.xml"> <!-- visualising semigroups -->
<#Include SYSTEM "z-chap20.xml"> <!-- utils -->
<!--<#Include SYSTEM "z-chap21.xml"> <!-- orbits -->
<!-- <#Include SYSTEM "z-chap22.xml"> <!-- extending the package -->
</Body>

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89 changes: 46 additions & 43 deletions doc/semieunit.xml
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Expand Up @@ -13,7 +13,7 @@
<Filt Name = "IsMcAlisterTripleSemigroup" Arg = "S"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This function returns true if <A>S</A> is a McAlister triple semigroup.
This function returns <K>true</K> if <A>S</A> is a McAlister triple semigroup.
A <E>McAlister triple semigroup</E> is a special representation of an
E-unitary inverse semigroup <Ref Oper="IsEUnitaryInverseSemigroup"/>
created by <Ref Oper="McAlisterTripleSemigroup"/>.
Expand All @@ -23,42 +23,39 @@

<#GAPDoc Label="McAlisterTripleSemigroup">
<ManSection>
<Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, Y, act"
Label = "for a group, partial order digraph, join-semilattice digraph, and action"/>
<Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, Y"
Label = "for a group, partial order digraph, and join-semilattice digraph"/>
<Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, sub_ver, act"
Label = "for a group, partial order digraph, homogeneous list, and action"/>
<Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, sub_ver"
Label = "for a group, partial order digraph, and homogeneous list"/>
<Oper Name = "McAlisterTripleSemigroup" Arg = "G, X, Y[, act]"/>
<Returns>A McAlister triple semigroup.</Returns>
<Description>

The following documentation covers the technical information needed to
create McAlister triple semigroups in GAP, the underlying theorey can be
read in the introduction of Section
<Ref Sect = "McAlister triple semigroups"/>.
create McAlister triple semigroups in GAP, the underlying theory can be
read in the introduction to Chapter
<Ref Chap = "McAlister triple semigroups"/>.
<P/>

In this implementation the partial orders <A>X</A> and <A>Y</A> of a
McAlister triple are represented by <Ref Oper="Digraph" BookName="Digraphs"/>
objects. The vertices of a directed graph represent the elements of a
partial order while the order relation is defined by <C>A</C> <M>\leq</M>
<C>B</C> if and only if there is an edge from <C>B</C> to <C>A</C>. The
digraph <A>Y</A> should be an induced subdigraph of <A>X</A> and the
<Ref Oper="DigraphVertexLabels" BookName="Digraphs"/> of <A>Y</A> must
correspond to the vertices of <A>X</A> it is induced from. This means that:
In this implementation the partial order <C>X</C> of a McAlister triple is
represented by a <Ref Oper="Digraph" BookName="Digraphs"/> object <A>X</A>.
The digraph represents a partial order in the sense that vertices are the
elements of the partial order and the order relation is defined by
<C>A</C> <M>\leq</M> <C>B</C> if and only if there is an edge from <C>B</C>
to <C>A</C>. The semilattice <C>Y</C> of the McAlister triple should be an
induced subdigraph <A>Y</A> of <A>X</A> and the
<Ref Oper="DigraphVertexLabels" BookName="Digraphs"/> must correspond to
the vertices of <A>X</A> on which <A>Y</A> is induced. That means that
the following:
<P/>

<C><A>Y</A> = InducedSubdigraph(<A>X</A>, DigraphVertexLabels(<A>Y</A>))
</C><P/>

must return true. Herein if we say that a vertex <C>A</C> of <A>X</A>
must return <K>true</K>. Herein if we say that a vertex <C>A</C> of <A>X</A>
is 'in' <A>Y</A> then we mean there is a vertex of <A>Y</A> whose label is
<C>A</C>. <P/>

<C>A</C>. Alerternatively the user may choose to give the argument
<A>Y</A> as the vertices of <A>X</A> on which <A>Y</A> is the induced
subdigraph. <P/>

A McAlister triple semigroup is created from a quadruple
<A>(G, X, Y, act)</A> where:
<C>(<A>G</A>, <A>X</A>, <A>Y</A>, <A>act</A>)</C> where:

<List>
<Item>
Expand All @@ -85,10 +82,10 @@
<C>A</C> in <A>X</A> and <C>g,h</C> <M>\in</M> <A>G</A>. Furthermore
the permutation represenation of this action must be a subgroup of the
automorphism group of <A>X</A>. That means we require the following
to return true: <P/>
to return <K>true</K>: <P/>
<C>IsSubgroup(AutomorphismGroup(</C><A>X</A><C>),
Image(ActionHomomorphism(</C><A>G</A><C>,
DigraphVertices(</C><A>X</A><C>), </C><A>act</A><C>))</C>. <P/>
DigraphVertices(</C><A>X</A><C>), </C><A>act</A><C>));</C> <P/>
Furthermore every vertex of <A>X</A> must be in the orbit of some
vertex of <A>X</A> which is in <A>Y</A>. Finally, <A>act</A> must fix
the vertex of <A>X</A> which is the minimal vertex of <A>Y</A>, i.e.
Expand All @@ -110,7 +107,8 @@ gap> x := Digraph([[1], [1, 2], [1, 2, 3], [1, 4], [1, 4, 5]]);
<digraph with 5 vertices, 11 edges>
gap> y := InducedSubdigraph(x, [1, 4, 5]);
<digraph with 3 vertices, 6 edges>
gap> SetDigraphVertexLabels(y, [1, 4, 5]);
gap> DigraphVertexLabels(y);
[ 1, 4, 5 ]
gap> A := AutomorphismGroup(x);
Group([ (2,4)(3,5) ])
gap> S := McAlisterTripleSemigroup(A, x, y, OnPoints);
Expand Down Expand Up @@ -143,8 +141,9 @@ gap> AsSemigroup(IsPartialPermSemigroup, T);
<Attr Name = "McAlisterTripleSemigroupPartialOrder" Arg = "S"/>
<Returns>A partial order digraph.</Returns>
<Description>
Returns the partial order digraph used to create the McAlister triple
semigroup <A>S</A> via <Ref Oper="McAlisterTripleSemigroup"/>.
Returns the <Ref Prop="IsPartialOrderDigraph" BookName="Digraphs"/> used
to create the McAlister triple semigroup <A>S</A> via
<Ref Oper="McAlisterTripleSemigroup"/>.
</Description>
</ManSection>
<#/GAPDoc>
Expand All @@ -154,16 +153,17 @@ gap> AsSemigroup(IsPartialPermSemigroup, T);
<Attr Name = "McAlisterTripleSemigroupSemilattice" Arg = "S"/>
<Returns>A join-semilattice digraph.</Returns>
<Description>
Returns the join-semilattice digraph used to create the McAlister triple
semigroup <A>S</A> via <Ref Oper="McAlisterTripleSemigroup"/>.
Returns the <Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/>
used to create the McAlister triple semigroup <A>S</A> via
<Ref Oper="McAlisterTripleSemigroup"/>.
</Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="McAlisterTripleSemigroupAction">
<ManSection>
<Attr Name = "McAlisterTripleSemigroupAction" Arg = "S"/>
<Returns>A group.</Returns>
<Returns>A function.</Returns>
<Description>
Returns the action used to create the McAlister triple semigroup
<A>S</A> via <Ref Oper="McAlisterTripleSemigroup"/>.
Expand All @@ -173,15 +173,16 @@ gap> AsSemigroup(IsPartialPermSemigroup, T);

<#GAPDoc Label="IsMcAlisterTripleSemigroupElement">
<ManSection>
<Prop Name = "IsMcAlisterTripleSemigroupElement" Arg = "x"/>
<Prop Name = "IsMTSE" Arg = "x"/>
<Filt Name = "IsMcAlisterTripleSemigroupElement" Arg = "x"/>
<Filt Name = "IsMTSE" Arg = "x"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
Returns true if <A>x</A> has been created by
<Ref Oper="McAlisterTripleSemigroupElement"/>. The functions <C>IsMTSE</C>
and <C>IsMcAlisterTripleSemigroupElement</C> are synonyms. The
mathematical description of these objects can be found in the
introduction to <Ref Sect = "McAlister triple semigroups"/>.
Returns <K>true</K> if <A>x</A> is an element of a McAlister triple
semigroup; in particular, this returns <K>true</K> if <A>x</A> has been
created by <Ref Oper="McAlisterTripleSemigroupElement"/>. The functions
<C>IsMTSE</C> and <C>IsMcAlisterTripleSemigroupElement</C> are synonyms.
The mathematical description of these objects can be found in the
introduction to Chapter <Ref Chap = "McAlister triple semigroups"/>.
</Description>
</ManSection>
<#/GAPDoc>
Expand Down Expand Up @@ -243,7 +244,7 @@ gap> a * b;
cover of <A>S</A> is a surjective idempotent separating homomorphism from
a finite semigroup satisfying <Ref Prop="IsEUnitaryInverseSemigroup"/>
to <A>S</A>. A semigroup homomorphism is said to be idempotent separating
if no two idempotents are mapped to the same element of in the image.
if no two idempotents are mapped to the same element of the image.
<Example><![CDATA[
gap> S := InverseSemigroup([PartialPermNC([1, 2], [2, 1]),
> PartialPermNC([1], [1])]);
Expand Down Expand Up @@ -272,7 +273,8 @@ true]]></Example>
<Ref Attr="McAlisterTripleSemigroupPartialOrder"/> satisfies
<Ref Prop="IsJoinSemilatticeDigraph" BookName="Digraphs"/>. McAlister
triple semigroups are a represenation of E-unitary inverse semigroups and
more can be read about them in <Ref Sect="McAlister triple semigroups"/>.
more can be read about them in Chapter
<Ref Chap="McAlister triple semigroups"/>.
<Example><![CDATA[
gap> S := InverseMonoid([PartialPermNC([1, 2], [1, 2]),
> PartialPermNC([1, 2, 3], [1, 2, 3]),
Expand All @@ -295,8 +297,9 @@ true]]></Example>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This function determines whether a given semigroup is an F-inverse
monoid. A semigroup is an F-inverse monoid when it is a monoid which
satisfies <Ref Prop="IsFInverseSemigroup"/>.
monoid. A semigroup is an F-inverse monoid when it satisfies
<Ref Filt="IsMonoid" BookName = "ref"/> and
<Ref Prop="IsFInverseSemigroup"/>.
</Description>
</ManSection>
<#/GAPDoc>
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